How to calculate the atomic packing factor of the hexagonal-close-packed?

Calculating the atomic packing factor (APF) of a hexagonal close-packed (HCP) structure involves understanding the arrangement of atoms within the unit cell. The APF is a measure of how efficiently the atoms are packed in a crystal structure, defined as the ratio of the volume occupied by the atoms to the total volume of the unit cell. Let’s break this down step by step.

Understanding the HCP Structure

The hexagonal close-packed structure consists of two layers of atoms arranged in a hexagonal pattern, with a third layer that fits into the gaps of the first two layers. In total, there are 6 atoms per unit cell in an HCP arrangement. The unit cell can be visualized as having a hexagonal base and a height that accommodates the layers of atoms.

Step-by-Step Calculation

• Volume of Atoms: Each atom can be approximated as a sphere. The volume \( V \) of a single atom (sphere) is given by the formula:

V = \frac{4}{3} \pi r^3

• For HCP, there are 6 atoms in the unit cell, so the total volume occupied by the atoms is:

Total Volume of Atoms = 6 \times \frac{4}{3} \pi r^3 = 8 \pi r^3

• Volume of the Unit Cell: The volume \( V_{cell} \) of the hexagonal unit cell can be calculated using the formula:

V_{cell} = \frac{3\sqrt{3}}{2} a^2 c

where \( a \) is the length of the sides of the hexagon and \( c \) is the height of the unit cell.

• Finding Relationships: In an HCP structure, the relationship between \( a \) and \( c \) is given by:

\( c = \sqrt{\frac{8}{3}} a \)

• Substituting this into the volume of the unit cell gives:

V_{cell} = \frac{3\sqrt{3}}{2} a^2 \left(\sqrt{\frac{8}{3}} a\right) = 2\sqrt{6} a^3

• Calculating the APF: Now, we can find the atomic packing factor by dividing the total volume of the atoms by the volume of the unit cell:

APF = \frac{Total Volume of Atoms}{V_{cell}} = \frac{8 \pi r^3}{2\sqrt{6} a^3}

• To express \( r \) in terms of \( a \), we note that in HCP, \( r = \frac{a}{\sqrt{12}} \). Substituting this back into the APF formula allows us to simplify further.

Final Result

After substituting and simplifying, the atomic packing factor for the hexagonal close-packed structure is found to be:

APF = \frac{8 \pi}{3\sqrt{2}} \approx 0.74

This indicates that approximately 74% of the volume of the HCP unit cell is occupied by atoms, making it one of the most efficient packing arrangements in crystalline solids.

Real-World Implications

The high packing efficiency of HCP structures is significant in materials science and solid-state physics, influencing properties such as density, strength, and thermal conductivity. Understanding these packing factors helps in the design of materials with desired characteristics for various applications.